When Equilibria Bifurcate: Statistical Resolution Limits for Learning Equilibrium Correspondences

Learning in games often targets one profile with small exploitability. We show that this can be statistically much easier than learning the complete equilibrium correspondence: in a contiguous family of two-player zero-sum games, a data-independent profile has residual $O(T^{-1/2})$, while every adaptive $T$-query procedure incurs order-one minimax Hausdorff error for the exact correspondence. We therefore interpret approximation tolerance as statistical branch resolution. At a specified base game $A_{\star }$, we define a branch-survival cost and prove that, at every positive first-order resolution, approximate-equilibrium correspondences converge uniformly to its sublevel sets. The cost equals the duality gap of the perturbation game restricted to the base optimal-strategy polytopes and the support function of a critical-contrast polytope affinely equivalent to the complete base correspondence. Under local alternatives $A_{\star }+H/\sqrt{T}$, known-variance Gaussian payoff observations, and deterministic full-support sampling allocations, we obtain a finite-sample risk decomposition with an asymptotically vanishing strategic discrepancy and a local Gaussian random-set minimax limit. This yields a subcritical–critical–supercritical resolution transition and an adaptive lower bound for honest exact-set inference. The statistical comparison is fixed-dimensional and local oracle; complete Hausdorff convergence is proved at positive first-order resolution, while at exact resolution only an outer limit is universal.